scholarly journals Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Ranchao Wu ◽  
Xiang Li
Keyword(s):  
Hopf Bifurcation ◽  
Projective Synchronization ◽  
Linear Feedback ◽  
Normal Form Theory ◽  
Dynamical Behaviors ◽  
Control Scheme ◽  
Stable Periodic ◽  
Form Theory ◽  
Synchronization Scheme ◽  
Bifurcation And Stability

A new Rössler-like system is constructed by the linear feedback control scheme in this paper. As well, it exhibits complex dynamical behaviors, such as bifurcation, chaos, and strange attractor. By virtue of the normal form theory, its Hopf bifurcation and stability are investigated in detail. Consequently, the stable periodic orbits are bifurcated. Furthermore, the anticontrol of Hopf circles is achieved between the new Rössler-like system and the original Rössler one via a modified projective synchronization scheme. As a result, a stable Hopf circle is created in the controlled Rössler system. The corresponding numerical simulations are presented, which agree with the theoretical analysis.

scholarly journals Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

Bifurcations of Time-Delay-Induced Multiple Transitions Between In-Phase and Anti-Phase Synchronizations in Neurons with Excitatory or Inhibitory Synapses

2019 ◽  
Vol 29 (11) ◽  
pp. 1950147 ◽  
Author(s):  
Li Li ◽  
Zhiguo Zhao ◽  
Huaguang Gu
Keyword(s):  
Time Delay ◽  
Inhibitory Synapses ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Subthreshold Oscillations ◽  
Dynamical Behaviors ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Time-delay-induced synchronous behaviors and synchronization transitions have been widely investigated for coupled neurons, and they play important roles for physiological functions. In the present study, time-delay-induced synchronized subthreshold oscillations were simulated, and the bifurcations underlying the synchronized behaviors were identified for a pair of coupled FitzHugh–Nagumo neurons. Multiple transitions between in-phase and anti-phase synchronizations induced by the time delay were simulated for the inhibitory and excitatory couplings. Subcritical or supercritical Hopf bifurcations and the stability of the Hopf-bifurcating periodic subthreshold oscillations were acquired using center manifold reduction and normal form theory. The in-phase or anti-phase synchronizations of the stable periodic subthreshold oscillations, which appear for multiple values of the time delay, were interpreted with the related eigenspace. The distributions of the different dynamical behaviors, including the synchronizations and bifurcations in the two-parameter plane of the time delay and coupling strength, were acquired for both types of synapses, and the different roles of the inhibitory and excitatory couplings on the synchronization transitions were compared.

scholarly journals Hopf Bifurcation Analysis and Chaos Control of a Chaotic System withoutilnikov Orbits

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Na Li ◽  
Wei Tan ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Chaos Control ◽  
Chaotic Oscillation ◽  
Delayed Feedback Control ◽  
Normal Form Theory ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
The Stability

This paper mainly investigates the dynamical behaviors of a chaotic system withoutilnikov orbits by the normal form theory. Both the stability of the equilibria and the existence of local Hopf bifurcation are proved in view of analyzing the associated characteristic equation. Meanwhile, the direction and the period of bifurcating periodic solutions are determined. Regarding the delay as a parameter, we discuss the effect of time delay on the dynamics of chaotic system with delayed feedback control. Finally, numerical simulations indicate that chaotic oscillation is converted into a steady state when the delay passes through a certain critical value.

scholarly journals Dynamical Analysis of a Computer Virus Model with Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Juan Liu ◽  
Carlo Bianca ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Characteristic Root ◽  
Computer Virus ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Virus Model ◽  
Bifurcation And Stability

An SIQR computer virus model with two delays is investigated in the present paper. The linear stability conditions are obtained by using characteristic root method and the developed asymptotic analysis shows the onset of a Hopf bifurcation occurs when the delay parameter reaches a critical value. Moreover the direction of the Hopf bifurcation and stability of the bifurcating period solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical investigations are carried out to show the feasibility of the theoretical results.

scholarly journals HOPF BIFURCATION AND CHAOS IN TABU LEARNING NEURON MODELS

2005 ◽  
Vol 15 (08) ◽  
pp. 2633-2642 ◽  
Author(s):  
CHUNGUANG LI ◽  
GUANRONG CHEN ◽  
XIAOFENG LIAO ◽  
JUEBANG YU
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic Behavior ◽  
External Input ◽  
Neuron Models ◽  
Normal Form Theory ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
Memory Decay ◽  
Form Theory ◽  
The Stability

In this paper, we consider the nonlinear dynamical behaviors of some tabu learning neuron models. We first consider a tabu learning single neuron model. By choosing the memory decay rate as a bifurcation parameter, we prove that Hopf bifurcation occurs in the neuron. The stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are determined by applying the normal form theory. We give a numerical example to verify the theoretical analysis. Then, we demonstrate the chaotic behavior in such a neuron with sinusoidal external input, via computer simulations. Finally, we study the chaotic behaviors in tabu learning two-neuron models, with linear and quadratic proximity functions respectively.

Hopy Bifurcation and Stability Analysis in a Predator-Prey Model with Distributed Delays

2014 ◽  
Vol 1049-1050 ◽  
pp. 1400-1402 ◽  
Author(s):  
Hong Bing Chen
Keyword(s):  
Hopf Bifurcation ◽  
Stage Structure ◽  
Distributed Delays ◽  
Stable Equilibrium Point ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Form Theory ◽  
Necessary And Sufficient ◽  
Bifurcation And Stability

In this paper, A mathematical model of two species with stage structure and distributed delays is investigated, the necessary and sufficient of the stable equilibrium point are studied. Further, by analyze the associated characteristic equation, it is founded that Hopf bifurcation occurs when τ crosses some critical value. The direction of Hopf bifurcation as well as stability of periodic solution are studied. Using the normal form theory and center manifold method.

scholarly journals Stability and Hopf Bifurcation for a Delayed Computer Virus Model with Antidote in Vulnerable System

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Zizhen Zhang ◽  
Yougang Wang ◽  
Massimiliano Ferrara
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Local Stability ◽  
Center Manifold ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Virus Model ◽  
Bifurcation And Stability

A delayed computer virus model with antidote in vulnerable system is investigated. Local stability of the endemic equilibrium and existence of Hopf bifurcation are discussed by analyzing the associated characteristic equation. Further, direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are presented to show consistency with the obtained results.

HOPF BIFURCATION AND STABILITY ANALYSIS FOR A STAGE-STRUCTURED SYSTEM

2010 ◽  
Vol 03 (01) ◽  
pp. 21-41 ◽  
Author(s):  
JUNLI LIU ◽  
TAILEI ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Structured System ◽  
Explicit Formulae ◽  
Bifurcation And Stability ◽  
Theoretical Results

In this paper, we considered a time-delay predator–prey system, in which the prey has two life stages, juvenile and mature. Delay was regarded as the bifurcation parameter, we analyzed the characteristic equation of the system at the positive equilibrium, stability of the positive equilibrium and existence of Hopf bifurcation with delay τ in the term of degree are investigated. The explicit formulae which determine the direction of the bifurcations, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. To verify our theoretical results, a numerical example is also included.

scholarly journals Hopf bifurcation of a delayed worm model with two latent periods

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Worm Model ◽  
Distribution Of Roots ◽  
Influential Parameters

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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